# Newton's law of universal gravitation **Newton's law of universal gravitation** states that every particle of matter attracts every other particle with a [[Force|force]] directed along the line joining them, proportional to the product of the two masses and inversely proportional to the square of the distance between them: `F = G·m1·m2/r^2`, where G is the [[Gravity|gravitational]] constant, 6.674×10⁻¹¹ N·m²/kg².[^up1-const] The word that carries the weight is *universal*. The same expression that describes an apple leaving a branch describes the [[Moon|Moon]] holding station around the [[Earth|Earth]] and the [[Earth]] around the [[Sun|Sun]], and until 1915 nothing was known that it could not describe.[^principia] In the microsim below the reader stands on Newton's mountain and fires a cannonball horizontally, turning a single knob: the muzzle speed. Below `sqrt(G·M/r)` the ball falls back onto the planet; at exactly that speed the trajectory closes into a circle; between it and `sqrt(2·G·M/r)` the ball runs an [[Ellipse|ellipse]] with the planet's centre at one focus; above it the path opens into a hyperbola and never returns — [[Escape_velocity|escape]]. The path is stepped live with a velocity-Verlet integrator rather than a Runge–Kutta one, so the energy readout `E = v^2/2 − G·M/r` wobbles about a constant instead of creeping upward over hundreds of orbits, and the sign of that number decides which shape appears.[^matlab-orbit] A laboratory preset reuses the same G for two lead spheres, so the constant that governs [[Planetary_system|planetary systems]] is seen producing a few nanonewtons on a bench. On the [[Physics]] flagship this article serves Part I — History, the section *Newton's cannonball: gravitation and orbits* (row P7): the point at which the spine stops describing motion and starts explaining it, and the law whose 43-arcsecond failure at Mercury's orbit later opened the door to [[General_relativity|general relativity]]. ## History The law was published in 1687 in Newton's *Philosophiæ Naturalis Principia Mathematica*, whose third book applies it to the [[Moon]], the planets, the tides and the shape of the Earth.[^principia] An inverse-square dependence had been guessed at by several people — Robert Hooke pressed the idea on Newton in correspondence in 1679–80, and the priority quarrel that followed lasted the rest of both men's lives. What Newton supplied was not the guess but the proof that an inverse-square [[Force|force]] toward a focus produces exactly the elliptical orbits [[Johannes_Kepler|Kepler]] had extracted from Tycho Brahe's observations, and, harder, that a spherical body attracts as though its whole [[Mass–energy_equivalence|mass]] sat at its centre.[^principia] The decisive argument is the Moon test. Newton compared the acceleration of a falling body at the Earth's surface with the acceleration needed to hold the Moon in its orbit, and found the ratio matched the inverse square of the ratio of the distances. The computation still works on modern data: at the Moon's mean distance of 3.844×10⁸ m the required centripetal acceleration is 2.72×10⁻³ m/s², which is 9.82 m/s² divided by (3.844×10⁸/6.371×10⁶)² = 3,641 (derived).[^up1-astro] The constant G itself was not measured until long after. In 1798 Henry Cavendish used a torsion balance built by John Michell to detect the attraction between lead spheres in a sealed room, and reported the mean [[Density|density]] of the Earth as 5.48 times that of water.[^cavendish] He never wrote G; the conversion is ours. His experiment is the ancestor of the sim's laboratory preset, and the reason a textbook can put a number in front of the symbol at all. ### Newton's "causes hitherto unknown" Newton was careful to say what he had not done. The preface to the first edition speaks of particles impelled toward one another "by some causes hitherto unknown", and in the General Scholium added to the second edition of 1713 he refused to supply a mechanism: he had not been able to deduce the cause of gravity from the phenomena, and *hypotheses non fingo* — I do not feign hypotheses.[^principia] The law was offered as a description that worked, not as an account of why. Contemporaries found this unsatisfying, and said so. Action at a distance across empty space, instantaneous and with no intervening medium, looked to Cartesian critics like the occult qualities natural philosophy had spent a century expelling, and Newton privately agreed that it was absurd that one body should act on another through a vacuum without mediation. The modern resolution is that the criticism was correct and the law was still right within its domain: [[General_relativity|general relativity]] replaces instantaneous attraction with curvature of [[Spacetime|spacetime]] propagating at the [[Speed_of_light|speed of light]], and recovers Newton's expression as the weak-field, slow-motion limit.[^einstein1916] The "cause hitherto unknown" turned out to be geometry. ## Modern form In modern notation the magnitude of the attraction between two point masses is `F = G·m1·m2/r^2`, with G = 6.674×10⁻¹¹ N·m²/kg² and r the separation of the centres.[^up1-grav][^up1-const] The force acts on both bodies, equal in size and opposite in direction, as [[Newton's_laws_of_motion|Newton's third law]] requires, so the lighter body accelerates more: the Earth pulls a falling apple exactly as hard as the apple pulls the Earth. The size of G is the most important fact about the law in practice. Two 1 kg lead spheres whose centres are 100 mm apart attract with 6.67×10⁻⁹ N (derived) — about the weight of a 0.7 microgram speck of dust. Gravity is by a wide margin the weakest known interaction: for two protons the attraction is smaller than the [[Coulomb's_law|Coulomb]] repulsion by roughly 10⁻³⁶ (derived).[^up1-grav] It dominates the universe only because it has no negative [[Electric_charge|charge]] to cancel against, so in a large enough body every particle's contribution adds. This is the equation the microsim computes, and the controls are deliberately minimal. The planet's mass M and radius R are fixed; the reader sets only the horizontal muzzle speed v₀ at the summit. Three qualitatively different fates — return, capture, escape — come out of one formula and one slider, with no extra rule deciding which is which. The readouts are the orbital shape, the periapsis and apoapsis distances, and the specific energy `E = v^2/2 − G·M/r`, negative for the bound cases, zero at the parabolic boundary and positive on the hyperbola.[^matlab-orbit] ## Bodies with spatial extent Newton's law is stated for point particles, and the planets are not points. The bridge is the shell theorem, proved in Book I of the *Principia*: a uniform spherical shell attracts an external body exactly as though all its mass sat at its centre, and exerts no net force on a body anywhere inside it.[^principia] A solid sphere is a nest of such shells, so a [[Star|star]] or planet of spherically symmetric density acts externally as a point mass — which is why the [[Orbit|orbital]] two-body problem is tractable, and why the microsim can treat its planet as a single attracting centre. Inside a uniform sphere the result inverts. Only the shells below the observer contribute, their mass grows as r³, and the force therefore grows as r³/r² = r: gravity falls linearly to zero at the centre. At the surface the two expressions agree, giving `g = G·M/R^2`. With the Earth's mass 5.97×10²⁴ kg and mean radius 6.371×10⁶ m this is 9.82 m/s² (derived), against the standard 9.81 m/s²; the discrepancy is real, and is due to the Earth's rotation and equatorial bulge rather than to the arithmetic.[^up1-astro] Where the body is not spherically symmetric the shortcut fails and the attraction must be integrated over the distribution. The residual terms are small but consequential: the Earth's oblateness makes satellite orbital planes precess, and the variation of the Moon's pull across the diameter of the Earth — a difference of forces, not the force itself — is what raises the [[Tide|tides]]. ## Vector form Force has direction, and the scalar statement hides it. As a vector, the force on body 1 from body 2 is `F12 = −G·m1·m2·rhat12/r^2`, with `rhat12` the unit vector from 2 to 1 and the minus sign making the force attractive — always inward, always along the line of centres, never sideways.[^up1-grav] That last property makes gravitation a central force, and it is why [[Angular_momentum|angular momentum]] about the attracting centre is conserved: a force with no lever arm exerts no [[Torque|torque]]. For more than two bodies the forces simply add: the net force on body i is the vector sum of the pulls of every other body, each computed as though the others were absent. Superposition holds because the equation is linear in each mass, and it is what lets an n-body [[Verlet_integration|Verlet]] step be written as a double loop over pairs. It fails in [[General_relativity|general relativity]], where the field carries energy and therefore gravitates. Conservation of angular momentum is visible in the sim without being plotted: on an eccentric ellipse the ball sweeps quickly near periapsis and slowly near apoapsis, covering equal areas in equal times, which is [[Kepler's_laws_of_planetary_motion|Kepler's second law]] recovered from the vector form rather than assumed.[^cline-twobody] ## Gravity field It is often convenient to divide out the test mass and speak of the field: `g = −G·M·rhat/r^2`, an acceleration assigned to each point of space, whose value multiplied by any mass placed there gives the force on it. The field picture is bookkeeping in Newtonian gravity — the law has no medium and no delay — but it becomes physically essential in [[General_relativity|general relativity]], and it is the same conceptual move that turns [[Coulomb's_law|Coulomb's law]] into the [[Electric_field|electric field]] and the [[Gravitational_field|gravitational field]] into a property of space. Because the force is central and depends only on r, it is conservative and derives from a [[Potential_energy|potential energy]] `U = −G·M·m/r`, zero at infinite separation and negative everywhere else. Adding this to the kinetic term gives the specific energy the sim plots; setting it to zero defines the escape condition `v_e = sqrt(2·G·M/r)`, or, against launch height h above a planet of surface gravity g and radius R, `v_e = sqrt(2·g·R^2/(h + R))`.[^trench-escape] At the Earth's surface this is 11.2 km/s (derived), exactly √2 times the 7.91 km/s of a circular orbit at the same radius — the ratio the reader crosses with one slider. The potential also explains why the energy readout is the right diagnostic. Because U is unbounded below, an integrator that leaks energy does not merely accumulate a small error: it moves the trajectory between qualitatively different regimes, turning an ellipse into a slow outward spiral. That is the failure mode the Verlet stepper avoids.[^matlab-orbit] ## Limitations The law is an approximation with a known domain: weak fields, speeds small compared with light, and no propagation delay. Inside that domain it remains the working tool of [[Orbit|orbital]] mechanics, geodesy and stellar dynamics. Outside it, three things break. The attraction is instantaneous, which is incompatible with [[Special_relativity|special relativity]]. Energy does not gravitate, so light should be unaffected. And the orbits predicted are exactly closed, so no perihelion motion should occur beyond that caused by other bodies. ### Observations conflicting with Newton's formula The first crack was Mercury. After accounting for the perturbations of all the other planets, the perihelion of Mercury's orbit still advances about 43 arcseconds per century more than Newtonian theory allows — a discrepancy identified by Urbain Le Verrier in 1859 and refined over the following decades.[^einstein1915] Le Verrier had found Neptune by this method twelve years earlier, so the natural response was to postulate another planet, Vulcan, inside Mercury's orbit. It was searched for and never found. Two further conflicts appeared once relativity supplied predictions to test. Newtonian gravity acting on a corpuscle of light gives a deflection at the [[Sun|solar]] limb half the size [[General_relativity|general relativity]] predicts; the 1919 eclipse expeditions measured the larger value.[^eddington1920] And a clock deep in a gravitational potential runs slow, an effect with no Newtonian counterpart, measured in a laboratory tower as a fractional frequency shift of 2.5×10⁻¹⁵.[^cline-redshift] ### Einstein's solution [[Albert_Einstein|Einstein]]'s [[Einstein_field_equations|field equations]], completed in 1915, replace the force with the curvature of [[Spacetime|spacetime]] produced by its energy and momentum content; free bodies follow [[Geodesic|geodesics]], and no force appears in the equation of motion.[^einstein1916] The theory recovers `F = G·m1·m2/r^2` for weak fields and slow motion, and the leading correction produces a perihelion advance. Einstein computed the number for Mercury and got 43 arcseconds per century, with no adjustable parameter.[^einstein1915] The relationship between the two theories is the model for how physics replaces a law it has outgrown. Newton's expression was not shown to be wrong; it was shown to be the first term of something larger, exact in a limit that covers essentially every engineering application. A spacecraft trajectory is still computed from the inverse square, with relativistic corrections added where accuracy demands, and the timing corrections in satellite [[Radio_navigation|radio navigation]] are of the same character. ## Extensions Beyond the relativistic correction, the law has been probed for departures at both ends of its range. At short distances, torsion-balance experiments have tested the inverse square down to fractions of a millimetre without finding a deviation, constraining proposals in which gravity leaks into extra dimensions. At galactic scales the rotation curves of spiral galaxies are flat where the visible mass predicts a Keplerian falloff, read either as evidence for [[Dark_matter|dark matter]] or, by a minority, as a modification of the force law at very low acceleration.[^up3-cosmo] A separate line of extension keeps the law and changes the source. In [[General_relativity|general relativity]] the source is the whole stress–energy content rather than mass alone, so [[Pressure|pressure]] and energy density gravitate; on cosmological scales this is what allows a component with negative pressure — [[Dark_energy|dark energy]] — to accelerate the expansion. None of these extensions has displaced the Newtonian expression in the regime the microsim occupies: within the accuracy of the plot, a cannonball fired from a mountain obeys the 1687 formula exactly. ## Solutions For two bodies the problem is solved in closed form. Changing to the centre-of-mass frame and the relative coordinate reduces it to a single particle of reduced mass μ = m₁·m₂/(m₁ + m₂) moving in a fixed central potential, and conservation of energy and angular momentum reduce that to a one-dimensional problem in r with an effective potential.[^cline-twobody] The orbits are the conic sections: ellipse, parabola or hyperbola according to whether the energy is negative, zero or positive, which is the classification the sim's slider walks through.[^idema-kepler] For three or more bodies no such general solution exists, and the trajectories are integrated numerically, which makes the choice of integrator part of the physics. A general-purpose adaptive solver such as `ode45` does not conserve energy on an orbit; tightening its relative tolerance from the default 10⁻³ to 10⁻⁵ slows the drift but does not stop it.[^matlab-orbit] A second-order symplectic stepper — velocity Verlet — beats fourth-order [[Runge–Kutta_methods|Runge–Kutta]] over long runs, because it conserves a nearby "shadow" energy exactly, so the plotted E oscillates with amplitude of order Δt² rather than wandering.[^mech-drift] That is why the sim steps with Verlet and why the energy trace is on screen. Over a handful of orbits any competent method looks fine; the difference appears after hundreds, when the Runge–Kutta orbit has visibly shrunk and the Verlet one has not. The lesson generalises beyond gravitation: for an undamped [[Conservation_of_energy|conservative]] system, a structure-preserving [[Symplectic_integrator|symplectic integrator]] matters more than a high-order one.[^mech-drift] ## See also - [[Newton's_cannonball]] - [[Escape_velocity]] - [[Orbit]] - [[Gravity]] - [[Isaac_Newton]] - [[Kepler's_laws_of_planetary_motion]] - [[Newton's_laws_of_motion]] - [[General_relativity]] ## References [^principia]: Newton, Isaac (1687). *Philosophiæ Naturalis Principia Mathematica*. London: Royal Society. Book I, Propositions 70–71 (the shell theorem: no net force inside a uniform spherical shell; an external body attracted as if the mass were at the centre); Book III (the law applied to the Moon, the planets and the tides); the Author's Preface to the first edition ("by some causes hitherto unknown"); the General Scholium added to the second edition (1713) for *hypotheses non fingo*. [^up1-grav]: Sanny, Jeff; Ling, Samuel, et al. (2016). *University Physics Volume 1*. OpenStax, CC BY. Chapter 13, "Gravitation", pp. 611–664 (page to pin) — the statement of the law, its vector form, superposition over many bodies, gravitational potential energy and the orbit classification. Portal Book 077, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1. The extraction of this book lost most displayed equations; the forms quoted here are the standard ones and should be verified against the PDF pages. [^up1-const]: *University Physics Volume 1* (2016), Appendix C, "Fundamental Constants", pp. 883–884 (page to pin), for G = 6.674×10⁻¹¹ N·m²/kg². Portal Book 077. The ratio of gravitational to electrostatic force between two protons quoted in the text, ≈10⁻³⁶, is derived from this value together with Coulomb's constant and is not printed in the book. [^up1-astro]: *University Physics Volume 1* (2016), Appendix D, "Astronomical Data", pp. 885–886 (page to pin), for the Earth's mass 5.97×10²⁴ kg, mean radius 6.371×10⁶ m and the Moon's mean orbital distance 3.844×10⁸ m. Portal Book 077. The surface field 9.82 m/s², the Moon-test ratio 3,641, the centripetal acceleration 2.72×10⁻³ m/s², the escape speed 11.2 km/s and the circular speed 7.91 km/s are all derived here from those data and are not printed in the book. [^cavendish]: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth." *Philosophical Transactions of the Royal Society of London* 88: 469–526. Cavendish reported a mean density of 5.48 times that of water and did not state a value for G; the modern constant is obtained from his result by conversion. [^matlab-orbit]: Downey, Allen B. (2021). *Physical Modeling in MATLAB*, version 4.0. Chapter on two-body simulation, pp. 159–160, for the orbital force law `F = −G·m1·m2/r^2` with the Sun held fixed, the energy terms `KE = m·v^2/2` and `PE = −G·m1·m2/r` (p. 159), and the finding that an `ode45` orbit does not conserve energy, that tightening RelTol from 10⁻³ to 10⁻⁵ slows the drift, and that `ode23` "works surprisingly well" (p. 160). Portal Book 085, https://open.umn.edu/opentextbooks/textbooks/physical-modeling-in-matlab [^mech-drift]: Sub-manual 02 (*Mechanics, Dynamics and Differential Equations*), §5.2 "Energy drift: orbits and symplectic stepping", drawing on Portal Book 085 pp. 159–160 and Portal Book 030 pp. 89, 99–100: the dimensionless Kepler orbit (GM = 1, r₀ = 1, v₀ = 1) stepped by Euler, RK4 and velocity Verlet side by side; Euler spirals outward, RK4 drifts slowly, Verlet stays bounded because it conserves a shadow energy whose oscillation is of order Δt². [^trench-escape]: Trench, William F. (2013). *Elementary Differential Equations with Boundary Value Problems*. §4.3, "Elementary mechanics", pp. 167–169, for the escape-velocity derivation: `v·dv/dy = −g·R^2/(y + R)^2` integrates to `v^2/2 = g·R^2/(y + R) + v0^2/2 − g·R^2/(h + R)`, giving `v_e = sqrt(2·g·R^2/(h + R))`. Portal Book 032, https://open.umn.edu/opentextbooks/textbooks/elementary-differential-equations-with-boundary-value-problems. The section notes that this variant requires the inverse-square force and that the constant-g model is valid only near the surface (pp. 160, 167). [^cline-twobody]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised 2nd ed. Chapter 11, "Conservative two-body central forces", pp. 269–308 (page to pin), for the reduction to a one-body problem with reduced mass, the effective potential, the conic-section orbits, the conservation of angular momentum for any central force, and the absence of a general closed-form solution for three or more bodies. Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^idema-kepler]: Idema, Timon (2018). *Mechanics and Relativity*. Part I, Classical mechanics, pp. 80–87 (page to pin), on Kepler's laws and their derivation from the inverse-square law. Portal Book 080, https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^cline-redshift]: Cline (2018), *Variational Principles in Classical Mechanics*, pp. 103–104, for the gravitational redshift and the Pound–Rebka measurement, quoted there as a fractional frequency shift Δf/f = 2.5×10⁻¹⁵. Portal Book 073. Sub-manual 02 §A notes that the printed photon frequency and tower height on p. 103 are damaged in extraction and that a 22.5 m height reproduces the quoted shift through g·h/c²; the shift itself is the reliable figure. [^einstein1915]: Einstein, Albert (1915). "Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie." *Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin)*: 831–839. The anomalous advance of Mercury's perihelion, about 43 arcseconds per century, had been identified by Urbain Le Verrier in 1859 and refined by later observers. [^eddington1920]: Dyson, F. W.; Eddington, A. S.; Davidson, C. (1920). "A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919." *Philosophical Transactions of the Royal Society of London A* 220: 291–333. [^einstein1916]: Einstein, Albert (1916). "Die Grundlage der allgemeinen Relativitätstheorie." *Annalen der Physik* 354 (7): 769–822. [^up3-cosmo]: Sanny, Jeff; Ling, Samuel, et al. (2016). *University Physics Volume 3*. OpenStax, CC BY. Chapter 11, "Particle Physics and Cosmology", pp. 493–540 (page to pin), for galactic rotation curves, dark matter and dark energy as the standard account of them. Portal Book 079, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3. As with Volume 1, the extraction of this book lost most displayed equations and worked numbers; nothing numerical is taken from it here. ## External links - [*University Physics Volume 1*](https://openstax.org/books/university-physics-volume-1) — OpenStax, CC BY; Chapter 13 is the gravitation chapter this page follows (Portal Book 077) - [*Variational Principles in Classical Mechanics*](https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics), Douglas Cline — Chapter 11 for the two-body central-force treatment (Portal Book 073) - [*Physical Modeling in MATLAB*](https://open.umn.edu/opentextbooks/textbooks/physical-modeling-in-matlab), Allen Downey — the orbit-integration chapter behind the energy readout (Portal Book 085) - Further historical and archival sites are listed in the Wikipedia pair's *External links*; none is reproduced here until its URL has been checked. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Newton's_law_of_universal_gravitation.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Newton's law of universal gravitation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Newton's_law_of_universal_gravitation.html" data-title="Newton's law of universal gravitation"></div> *Built from `MICROSIM_GUIDE/specs/sims/Newton's_law_of_universal_gravitation.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation) : [Wikitube](https://en.wikitube.io/wiki/Newton's_law_of_universal_gravitation) · pinned revision [1370960529](https://en.wikipedia.org/w/index.php?oldid=1370960529) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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